Find the image of a point having the position vector: `3hati - 2hatj + hat k` in the plane `vec r.(3hati - hat j + 4hatk) = 2`
Solution
Let B be the root of point `A(3hati - 2hatj + hatk)` in the plane `vec r(3hati - hatj + 4hatj) = 2`can of AB: is
`barr(3hati - 2hatj + hatk) + lambda(3hati - hatj + 4hatk)`
`:. (x - 3)/3 = (y + 2)/(-1) = (z-1)/4 = lambda`
`:. x = 3lambda + 3`, `y = -lambda -2`, `z = 4lambda + 1`
subtitute x,y and z in plane 3x - y + 4z = 2
`∴ 3(3lambda + 3)-(-lambda - 2) + 4(4lambda + 1) = 2`
`9lambda + 9 + lambda + 2 + 16lambda + 4 = 2`
`26lambda + 13 = 0 => lambda = -1/2`
`:. x = -3/2 + 3`, `y = 1/2 - 2` `z = -2 + 1`
x = 3/2, `y = -3/2`, z = -1
∴ by midpt formula
`3/2 = (3+x_1)/2, (-3)/2 = (-2+y1)/z, -1 = (1-z_1)/2`
`x_1 = 0` `y_1 = -1`, `-z = 1 - z_1`
`z_1 = 1+ 2 = 3`
